Question

A non-relativistic particle of mass m is held in a circular orbit around the origin by an attractive force f (r) = −kr where k is a positive constant 

(a) Show that the potential energy can be written 

U(r) = kr²/2 

Assuming U(r) = 0 when r = 0 

(b) Assuming the Bohr quantization of the angular momentum of the particle, show that the radius r of the orbit of the particle and speed v of the particle can be written

where n is an integer 

(c) Hence, show that the total energy of the particle is

(d) If m = 3 × 10⁻²⁶ kg and k = 1180 N m⁻¹, determine the wavelength of the photon in nm which will cause a transition between successive energy levels.

Answer 1

(b) Bohr’s assumption of quantization of angular momentum gives 

mvr = nℏ (1) 

Equating the attracting force to the centripetal force.